Stability Analysis of a Current Source Rectifier based Drive for Aerospace Applications

Stephen J Forrest and Jiabin Wang Department of Electronic and Electrical Engineering, the University of Sheffield

Mappin Street, Sheffield, S1 3JD, United Kingdom

Abstract – This paper presents a theoretical analysis of the criteria which govern the stability of a current source rectifier drive, for a switched reluctance machine or actuator. A small signal stability analytical model, employing a state-space averaging technique, is established and verified by a comparison with a time-domain numerical simulation study.

In order to fulfil the power quality requirements of the ac bus, a passive input filter is employed in the drive, which together with the supply impedance may result in unstable operation, since a power electronic converter behaving as a constant power load exhibits negative incremental input impedance. Given the array of valid filter parameters, a seemingly well designed input filter which satisfies the power quality requirements, can often cause significant performance and control degradation when installed in to a closed-loop controlled switching converter.

Index Terms – Constant power load (CPL), current source rectifier, motor drives, negative impedance instability, small-signal stability analysis, power converters, switched reluctance machine.

I. INTRODUCTION

A direct converter topology for connecting an N-phase Switched Reluctance (SR) Machine or Actuator to a 3- phase ac supply is shown in Fig. 1. The lack of reactive components in the dc-link is one of the salient advantages which allow a compact, yet highly reliable, drive to be realised. In particular, for an aerospace application, the limitation of absorbed reactive power on to the ac bus, strict power quality requirements and requirements for controllable input power factor, minimal energy storage and flexibility in terms of independent machine phases, make the current source rectifier (CSR) an ideal converter topology for this application. The need for only filtering level capacitance allows other capacitor technologies with superior reliability to be considered, such as plastic-film dielectric (polypropylene or polyester) and multilayer capacitors. Although relatively large in size and weight, plastic film dielectric capacitors are well suited to ac applications and offer tight tolerances with very little change in capacitance with variation in temperature. Their low energy dissipation factor and high capacitance permits high ac currents and offers good reliability. It is well recognised that adopting more electric technologies on modern aircraft is reliant upon realising power electronic converters with both the desired volumetric power density and required reliability for safety critical applications, with minimal distortion on supply-side power quality.

For an idealised case, when the sum of the net winding current for N-output phases is held at a constant value for a

given output power, the converter behaves as a current source rectifier/inverter. An input filter is employed to reduce the input current and input voltage distortion which occurs due to harmonics introduced by modulating these currents and voltages. Thus, three sinusoidal input-phase currents, with the required power factor and power quality can be synthesised at the supply terminals, although the currents drawn by the individual phase modules will be far from sinusoidal.

II. STABILITY ANALYSIS

Closely controlled loads, such as the current source converter can act as constant power loads (CPL) to the ac supply, having negative incremental input impedance within the frequency range of the system control loops. The presence of the input filter together with the supply impedance or cable effect affects the transfer functions of the converter system and as a consequence, negative incremental input resistance may leave the system susceptible to further degradation of power quality and system stability at certain operating points [1-3]. Inappropriate filter parameters can lead to unstable operation, which becomes particularly significant as the closed-loop current control bandwidth is increased. 1

1 This work was supported in part by Rolls-Royce PLC

Fig. 1. Low energy storage converter topology for controlling bi-directional power flow between an N-phase SR machine or Actuator,

and a 3-phase ac supply.

978-1-4673-4974-1/13/$31.00 ©2013 IEEE911

Given the array of valid filter parameters, a seemingly well designed input filter, which satisfies the power quality requirements, can often cause significant performance and control degradation when installed in to a closed-loop controlled switching converter. Insufficient damping of the input filter results in a significant peak output impedance at its resonant frequency. Furthermore, due to the number of possible combinations of filter topologies and parameters, a large variety of dynamic interactions are possible. The complex interaction between the filter and the converter can be analysed by considering the transfer function of the converter, input filter and supply.

Small-signal stability analysis has previously been employed in studying the stability of power systems and matrix converters [4, 5]. This paper presents small signal stability analysis for a single current source rectifier (CSR), from a converter driving an N-phase SR machine or actuator, by employing a state-space averaging technique. Dynamic models of the input filter and a single phase load have been established, and the characteristics of the current source rectifier have been derived for a space vector modulation based current controller. Eigenvalue analysis is used to reveal the quantitative information of the stability modes for the CSR system under a given operating condition and for a given set of filter parameters.

III. MODELLING OF THE CURRENT SOURCE RECTIFIER SYSTEM

A schematic of the system is shown in Fig. 2 and comprises of a non-ideal ac supply, input filter stage and a current source rectifier operating under current feedback control at constant power. A parallel damped LLC filter is employed, where Lf and Cf are the filter inductance and capacitance respectively and Rfd and Lfd are the respective resistance and inductance of the parallel damping branch. The effect of the non ideal three phase AC source and supply cables are represented as a combined impedance by inductance Ls and resistance, Rs. Thus, the dynamic

behaviour of the supply filter and current source rectifier is established as follows:

According to the space vector based modulation scheme, the average output voltage of the current source rectifier, over a switching period, is given by (1). As the peak ac voltage increases or decreases, the modulation index m is adjusted by the PWM controller to maintain a constant output voltage and dc load current. Under these conditions the CSR is insensitive to any input voltage variation [2]. 32 · (1)

· (2)

· (3)

Equation (2) represents the voltage transfer relationship of the current source rectifier. It can be seen that if the peak voltage used in the modulation Ui, is equal to the peak converter input voltage Vim, then and a perfect voltage transfer is achieved. Equation (3) represents the current transfer relationship of the current source converter. When the above operating conditions are met, the lossless converter represents a constant power load to the ac source.

IV. LOAD SIDE DYNAMIC EQUATIONS AND CURRENT CONTROL

A. SRM Winding

A simple equivalent circuit for the SRM winding can be derived as follows. Mutual coupling between the machine phases is usually zero or very small and has therefore been neglected [6]. The terminal voltage of a winding is therefore equal to the sum of the resistive voltage drop and the rate of change of the flux linkages (4), where Idc and Vdc are the net winding current and voltage respectively, Rd is the resistance per phase, θSR is the angular position of the SR rotor and ψSR is the flux linkage per phase [7].

115V, 400HzThree Phase

Supply

Cf

Lfd Rfd

Lf

Ls RsEd

Rd

Ld

S1 S3 S5

S4 S6 S2

D1 D3 D5

D4 D6

D2

Vdc

Idc

Ia Iai

ICf

Digital ControllerVoltage Feedback

Current Feedback

Input Filter

Fig. 2. Current Source Rectifier System.

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· , (4)

The flux linkage, ψSR, may be expressed as (5), where Ld is the rotor position and phase current dependent machine phase inductance. Due to the doubly salient construction of the SR motor (i.e. both the rotor and the stator have salient poles) magnetic saturation effects are significant and in general, the flux linkage in an SRM phase varies as a function of rotor position, θSR, and the motor current, Idc. , · (5)

For the switched reluctance machine, induced electromotive force and current are not only relative with static (apparent) inductances, but also depend on dynamic (incremental) inductances, hence,

The first term in (7) is referred to as incremental inductance. The sum of the two terms in (7) is defined as Ld. Without loss of generality, the instantaneous dc voltage applied to the SRM phase winding can be represented as: , , · · (8)

Further manipulation gives the expression in (8) where the three terms on the right hand side represent the resistive voltage drop, inductive voltage drop and induced emf, respectively, where ωm is the mechanical rotor speed. The induced emf, Ed, can therefore be expressed as:

where, Ke, is the emf coefficient, and is obtained with a constant current at a selected operating point. Assuming that the magnetizing characteristic of the core is linear, and mechanical power is given as Pm = ωmTe , the instantaneous electromagnetic torque per a phase, Te is expressed as follows: 12 · ,

(11)

A representative SRM winding on the load side of the system can therefore be modelled by the following differential equation:

(12)

B. Current Control

In order to control the current in the phase winding, a proportional plus integral (PI) controller is used. The output voltage reference of the current controller, , can be expressed as (13), where the proportional and integral gains are represented by Kp and Ki respectively.

(13)

Simplification yields the expression in (14), where represents the internal state of the d-

axis PI current controller.

(14)

By substituting (2) and (14) into (12), the governing equations for the current in a single phase winding of an SR machine can be derived as follows: 1

1

(15)

(16)

For a current source rectifier operating such that the net dc load current is controlled to be constant, the average power at the input to the converter is assumed to be constant, since the input voltage is constant and the active input current is linearly proportional to the dc current. The relationship between the input and output current and the input and output voltage of the converter can therefore be established, where the following assumptions are made:

(i) The effect of switching harmonics are neglected. (ii) The averaged-value modelling technique is used, for the

nonlinear part of the system, thus only average current and voltage are considered over a switching period.

(iii) The power electronic devices are ideal, thus IGBT switching and conduction losses and diode conduction losses are neglected.

Naturally the validity of assumptions (i) and (ii) improves as switching frequency increases. Under these assumptions, the output voltage and the input current of the current source rectifier under space vector modulation can be expressed as (17) and (21) respectively, where Vim is the peak ac voltage across the capacitor Cf or peak voltage at the input to the current source rectifier.

Ui represents the peak voltage that the controller assumes to be applied to the load, and thus is the peak voltage used by the modulation scheme to calculate the modulation index.

,, ·

(6)

where: , , ·,

(7)

, · · · · (9)

, (10)

· (17)

(18)

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Vdi and Vqi are the voltages at the input to the converter, expressed in the ds-axis and qs-axis reference frame which rotates at the angular frequency ωi of the ac supply. If the peak voltage used in the modulation is exactly equal to the peak supply voltage, a perfect voltage transfer is achieved and thus . Equating instantaneous input power to the instantaneous output power [8], for a CSR, gives the following: · 32 (19)

where Idi and Iqi are the ds- and qs- axis components of the converter input current. If unity power factor is not required, then the reactive power at the input to the converter is zero, i.e: 32 0 (20)

Solving for Idi and Iqi in (19) and (20) yields expressions for the input currents (21), where Pdc is the power drawn by the current source rectifier. · 23 · 23 · · ·23 · ·

(21) · 23 · 23 · · ·23 · ·

It should be noted that the behaviour of the system will depend on the modulation strategy however, under the condition where perfect voltage modulation is achieved the current source rectifier becomes insensitive to any input voltage variation and behaves as a constant power load. In this case, the stability of the system is only dependent on the power drawn by the CSR and the chosen combination of filter parameters. In reality however, practical implementation issues prevent perfect voltage transfer and the actual average applied to the load over a switching period, does not equal the voltage reference of the current controller. Thus, the stability of the system will depend on the dynamic behaviour of the current source rectifier and consequent interaction with the input filter.

V. ANALYSIS OF INPUT FILTER

Fig. 3. Per phase equivalent circuit of the input filter.

A per phase equivalent circuit of the third order parallel Rfd-Lfd damped LLC input filter is shown in Fig. 3. The dynamic equations of this input filter topology can be derived from the appropriate circuit laws and are given as follows:

VI. STATE SPACE ANALYSIS OF THE SYSTEM

From (22) the nonlinear state space equations of the system can be derived in the synchronous reference frame, which are expressed by (23), where IdL and IqL, are the ds-axis and qs-axis current components in the primary inductor Lf, Idfd and Iqfd are the ds-axis and qs-axis current components in the filter damping branch Lfd, and Vds and Vqs are the ds-axis and qs-axis components of the supply voltage. It should be noted that for the filter topology shown in Fig. 3, additional Ls and Rs components have to be introduced in order to represent the effects of the cable impedance.

(23)

1 1

1 1

(22)

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Combining the dynamic equations of the load and controller (15) and (16) with the equations of the input filter (23), yields the state space equations for the current source rectifier, load and input filter inclusive (24) to (27), where , is the vector of the non-linear functions of X and Y. The vectors of the state variables X and inputs Y of the CSR system are given by (26) and respectively, , (24)

(25)

, (26)

where Vm is the magnitude of the ac supply voltage,

.

1 1

(27)

1 11 23

1 11 23

VII. STABILITY OF THE CURRENT SOURCE RECTIFIER

The stability of the CSR system can be evaluated by computing the eigenvalues of the Jacobian matrix of the state space equations (24) [9] under a given operating condition. The influence of input filter parameters and output power level on system stability can be studied accordingly.

Fig. 4. Phasor diagram of phase A voltage and synchrous reference frame at the converter ac terminals.

In order to simplify the computation, the ds-axis of the synchronously rotating reference is aligned with the peak phase A voltage at the input to the converter, Fig. 4. Thus, Vqi0 = 0 and Vdi0 = Vim0, where Vim0, Vdi0 and Vqi0 are the peak, d-axis and q-axis voltages at the converter terminals,

respectively, i.e., .

Fig. 5. Current source rectifier equivalent circuit in steady state.

The per phase steady state equivalent circuit diagram of the input filter and ideal constant power load, represented by a negative dynamic resistance (28), is shown in Fig. 5 The values of voltage Vim0,, the gamma angle and the currents in the filter inductances at a given operating point can be determined iteratively. 32 (28)

A. Power Factor Correction

It has been shown that by introducing a phase shift between the synchronous modulation reference and the supply-voltage based reference frame, it is possible to compensate for the leading power factor which results from the capacitive currents drawn by the supply filter. The theory of the reactive power compensation method is detailed in [10, 11] however, by expressing the supply filter equations in the synchronous rotating reference frame, the appropriate phase shift correction angle can be calculated. With reference to Fig. 5, solving for ICf and considering the direct and quadrature components in the supply-voltage

915

based reference frame, a simplified expression for reactive power compensation can be derived as follows.

(29)

Fig. 6. Reactive power compensation in supply based reference frame.

The reactive current drawn by the capacitor is given by (29). Therefore, in order to compensate for the reactive power, the converter needs to compensate by an amount equal to , which corresponds to the reactive power of the converter (30).

(30)

A phasor diagram of the reactive power compensation scheme, in the supply based reference frame is shown in Fig. 6, where the reference axis is selected to align with the peak of phase A voltage at the converter input. Angle, α represents the required phase shift angle in order to achieve unity power factor. Combining the active power balance equation with the reactive compensation equation, yields expressions for the converter input current. 2 3 ·

23

(31) 2 3 ·23

The equivalent circuit for calculating the initial condition of the converter system in steady state, with the implementation of reactive power compensation, is shown in Fig. 7. It should be noted that the capacitor is removed from the calculation when unity power factor is achieved, which is indicated by broken lines in the diagram. Thus, the steady state value of converter terminal voltage Vim0 should be calculated differently once unity power factor has been achieved. For the purposes of the analysis, the peak voltage used in the modulation, Ui, is assumed to be the nominal supply voltage.

Fig. 7. Steady state equivalent circuit, when reactive power compensation is implemented.

VIII. INFLUENCE OF FILTER PARAMETERS, CONTROL BANDWIDTH, AND OUTPUT POWER ON STABILITY

A. Varying Filter damping with Fixed Output Power

It can be shown that, at frequency ωn, the output impedance of the filter when Rfd tends to infinity, is equal to the output impedance of the filter when Rfd tends to zero. Thus the output impedance of the filter becomes independent of Rfd and optimal damping occurs for the choice of Rfd that causes the peak output impedance to occur at frequency ωn [12]. Fig. 8 shows the loci of the dominant pair eigenvalues which result with the parallel damped LLC filter when the current source rectifier is operated under the conditions given in TABLE I, with current control loop bandwidth, fcl = 800Hz, where Rfd is varied between 0.001Ω and 75Ω. It can be seen that system is less stable as the value of Rfd tends to zero or becomes large and the system becomes unstable when the damping resistor is 60Ω and above or less than 0.001Ω. Similarly the influence of the filter inductor and capacitor on system stability can be equally investigated.

Lfd

ReCf

Rfd

Lf

Is

Ic

Vim0

Ii

Vs

Ifd

ILf

TABLE I CSR FILTER PARAMETERS AND OPERATING CONDITIONS

Vs 115Vrms Ld 6.57mH Lf 1.31mH Rd 0.188Ω Lfd 1.01mH Ed 100V Cf 10.0μF ωs 400Hz

Fig. 8. Loci of eigenvalues for current source rectifier and filter with fixed Lf, Cf and Lfd with varying damping resistance, fcl = 800Hz. Pdc = 3kW.

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Rfd = 5.0 Ω

Rfd = 10.0 Ω

Rfd = 30.0 Ω

Rfd = 50.0 Ω

Rfd = 60.0 Ω

Rfd = 75.0 Ω

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Fig. 9. Loci of eigenvalues for the Current Source Rectifier and filter with parameters Lf, Cf and Lfd as in TABLE II, varying current control

bandwidth, where Rfd = 75Ω and Pdc = 3kW.

Fig. 10. Loci of Eigenvalues pairs for Current Source Rectifier with fixed filter parameters and varying Pdc, where fcl = 1000Hz and Rfd = 15Ω

Fig. 11. Loci of Eigenvalues of the CSR and filter operating under the conditions given in TABLE II. For = 17A the system shows stable

behaviour and for = 20A the system shows unstable behaviour.

Fig. 9 shows the loci of the two dominant pair eigenvalues which result as the current controller bandwidth is varied between 200Hz and 1 kHz, when the current source rectifier is operated under the conditions given in TABLE I. It can be seen that the stability margin is significantly influenced by control bandwidth and decreases as fcl increases. For fcl = 800Hz, the eigenvalues exist close to the stability margin, however for fcl = 1000Hz the eigenvalues

exist in the right half of the s-plane and the system becomes unstable. As fcl increases the drive behaves much closer to a constant power load, due to the improved current control, hence the decrease in stability with increased current control bandwidth.

B. Varying Output Power with Fixed Filter Parameters

Fig. 10 shows the variation of the dominant eigenvalues pairs which result with the filter parameters and CSR operating conditions given in TABLE I, for increasing output power and with a fixed emf, Ed. These are the parameters for which the experimental converter is designed and it can be seen that the converter remains stable over the intended 3.5kW operating range.

Fig. 11 shows an analytically predicted stable and unstable condition for the CSR operating under the conditions given in TABLE II.

IX. VALIDATION WITH TIME DOMAIN SIMULATIONS

In order to validate the eigenvalue analysis, time domain simulations were undertaken using MATLAB Simulink with SimPowerSystems blockset. Ideal power electronic switches and diodes are used to model the current source inverter power stage, with a switching frequency of 9600Hz. The bandwidth of the closed-loop current controller was set to 800Hz. The following section therefore presents a comparison of an analytically predicated unstable condition and the corresponding time domain simulation for this condition. It should be noted that the reactive power is controlled such that unity power factor is achieved at the supply side. It will be seen that following a 3A step increase in load current, the CSR exhibits unstable behaviour and growing oscillations can be seen in the output current.

Fig. 12. Simulated step change in load current from 17A to 20A, with power factor correction implemented.

TABLE II CSR OPERATING CONDITIONS AND INPUT FILTER PARAMETERS

Lf 1.31mH Rs 0.1Ω Lfd 1.01mH Ls 700μH Cf 10.0μF Ld 6.57mH Rfd 100Ω Rd 0.188Ω fcl 800Hz Ed 90V Vs 115Vrms Idc (initial) 17A Power factor correction angle, α 0.59 rads

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fcl = 200 Hz

fcl = 400 Hz

fcl = 600 Hz

fcl = 800 Hz

fcl = 1.0 kHz

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Fig. 13. Simulated peak supply side three phase line currents, with power factor correction implemented.

Fig. 14.Simulated peak three phase capacitor voltages, Vi, with power factor correction implemented.

The simulation parameters, converter operating conditions and input filter parameters are given in TABLE II. Figs. 12-14 show the load current, the supply side input current and the voltage at the input to the rectifier respectively, whilst the CSR is operating with constant output power and reactive power correction implemented, where a step increase of 3A is applied at 0.2 secs. It can be seen that the system becomes unstable as a result of the step change in load current from 17A to 20A. This result coincides with the analytically predicted unstable condition shown in Fig. 11. It can be seen that the instability is manifested as increasing high frequency oscillations in the dc current. Consequently, the input current and the voltage across the filter capacitor are distorted and become excessively high when the instability occurs. It should be noted, however, currents and voltages are naturally limited due to limitations on the modulation index within the controller and impedances in the circuit within a practical system. This condition could lead to catastrophic failure of the converter hardware.

X. SUMMARY

An analytical approach for predicting the stability of the current source rectifier and parallel damped LLC filter has been established, where a state space average model of the whole system has been developed. A complete evaluation of the stability of current source rectifier has been carried

out using a small-signal analysis around a steady-state operating point. The stability of the system has been evaluated by analysing the migration of eigenvalues of the linearised state matrix. The instability phenomena arise as persistent oscillations superimposed on the fundamental components of both ac voltage and current and dc output current. It has been demonstrated that the control bandwidth, input filter damping factor and active power drawn by the converter have a significant influence on the system stability. This has allowed relationships that give the stability of the system as a function of maximum output power, controller bandwidth and input filter damping factor by demonstrating that the system may become unstable as the output power exceeds a limit value. It has been seen that as the control bandwidth increases, input power increases and input filter damping factor decreases, the stability margin decreases. Time domain simulations have been performed to verify the proposed analytical stability model with and without reactive power compensation. A good correlation been observed in both cases.

REFERENCES

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[3] M. Molinas, D. Moltoni, G. Fascendini, J. A. Suul, and T. Undeland, "Constant power loads in AC distribution systems: An investigation of stability," in Industrial Electronics, 2008. ISIE 2008. IEEE International Symposium on, 2008, pp. 1531-1536.

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[5] J. Wang and M. Bouazdia, "Stability of matrix converter-fed permanent magnet brushless motor drive systems," in Power Electronics and Applications, 2009. EPE '09. 13th European Conference on, 2009, pp. 1-10.

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[7] R. Krishnan, Switched reluctance motor drives: modeling, simulation, analysis, design: CRC Press LLC, 2001.

[8] L. Kyo-Beum and F. Blaabjerg, "Simple Power Control for Sensorless Induction Motor Drives Fed by a Matrix Converter," Energy Conversion, IEEE Transactions on, vol. 23, pp. 781-788, 2008.

[9] R. Vargas, U. Ammann, B. Hudoffsky, J. Rodriguez, and P. Wheeler, "Predictive Torque Control of an Induction Machine Fed by a Matrix Converter With Reactive Input Power Control," Power Electronics, IEEE Transactions on, vol. 25, pp. 1426-1438.

[10] M. Salo and H. Tuusa, "vector-controlled PWM current-source-inverter-fed induction motor drive with a new stator current control method," Industrial Electronics, IEEE Transactions on, vol. 52, pp. 523-531, 2005.

[11] S. J. Forrest, J. Wang, G. W. Jewell, C. M. Johnson, and S. D. Calverley, "Analysis of an AC fed direct converter for a switched reluctance machine in aerospace applications," in Power Electronics and Motion Control Conference, 2006. IPEMC 2006. CES/IEEE 5th International, 2006, pp. 1-6.

[12] R. D. Middlebrook, "Design techniques for preventing input-filter oscillations in switched-mode regulators," presented at the Proceedings of Powercon 5, 1978.

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